Eventual n-periodicity of maximum-independent-set counts

For positive integers $n$ and nonnegative integers $d$, let $G_n(d)$ be the graph whose vertices are the monomials of degree $d$ in $n$ variables, with two monomials adjacent exactly when their least common multiple has degree $d+1$. Let $\mathbf{a}_n(d)$ denote the number of maximum independent sets of $G_n(d)$. Periodicity conjecture. For any $n \geq 1$, there exists some $N \geq 0$ such that the sequence $\{\mathbf{a}_n(d)\}_{d \geq N}$ is $n$-periodic.

This is the paper's second conjecture concerning the eventual behavior of maximum-independent-set counts. It is posed alongside the conjecture on eventual uniqueness at degrees divisible by $n$; the supplied text gives no general proof or resolution.